6.889 Sublinear Time Algorithms February 25, Lecture 6

Transcription

1 6.9 Sublinear Time Algorithms February 5, 019 Lecture 6 Lecturer: Ronitt Rubinfeld Scribe: Michal Shlapentokh-Rothman 1 Outline Today, we will discuss a general framework for testing minor-free properties by looking at a way to test planarity. Problem Setup and Definitions Algorithm Given Partition Oracle P Implementing Partition Oracle P Problem Setup and Definitions Our overall goal is to be able to distinguish graphs with a property P from all other graphs in sublinear time. However, there is no (known) method to do this exactly without exploring the whole graph. So, we will make a compromise: can we distinguish graphs with property P from graphs that are ɛ-far away from having property P? We will show how to distinguish such graphs by looking at the property of All graphs Graphs that are ɛ-close Graphs with Property to P property P Figure 1: We want to distinguish the line between the blue circle and the red circle planarity. First, we will start with several definitions. Note that all graphs have max degree of d. Definition 1 H is a minor of a graph G if we can obtain H from G via vertex removals, edge removals, or edge contractions. Figure : The graph on the left, is a minor of the graph on the right 1

2 Definition G is H minor free if H is not minor of G Definition 3 G is ɛ-close to H minor free if can remove ɛdn edges to make it H minor free. (Otherwise, G is ɛ-far) Definition 4 For a minor close property P, if G P, then all minors of G are in P Our goal is to test H minor freeness which means we want to pass H minor free graphs and fail graphs that are ɛ-far from H minor free. The following theorem and definitions will be helpful. Theorem 5 (Robertson and Seymour) Every minor-closed property is expressible as a constant number of excluded minors. Examples of minor-closed properties include K 3,3 or K 5, non-planar graphs and bounded tree width. Definition 6 G is (ɛ, k)-hyperfinite if we can remove ɛn edges and remain with connected components of size k. In other words, we have a (ɛ, k)-hyperfinite graph if we can remove a few edges and break up graph into small components. We can connect the notion of hyperfiniteness to H minor freeness. Theorem 7 Given H, C H such that 0 < ɛ < 1, every H minor free of graph of degree d is (ɛd, C H ɛ )-hyperfinite. All of this means that subgraphs of H minor free graphs are also H minor free and also hyperfinite. Why is hyperfinitiness useful? We can use it to prove that a graph is H minor free. We can partition a graph G into G by removing only a few edges. If non-constant size components remain, then we know that G is not H minor free. What if G is ɛ-close to having the property? Then we know that G is ɛ-close to having the property. Furthermore, it is particularly easy to test whether G has the property since we can just pick random components and in constant time test if they have the property. What we need now is is a local (sublinear) way to determine G. First, we will assume that we have a partition oracle P and then we will construct the partition oracle. 3 Algorithm Given Partition Oracle Our partition oracle will work as follows: Input: vertex v Output: p[v] (where v is the partition) name such that v V 1. P [v] k. P (v) is connected If G is H minor free, then with probability 9 10, u, v E P [u] + P [v] ɛd 4 n. In the first half of the algorithm, we are determining if the partition oracle looks right meaning are there few crossing edges and in the second part we are testing random partitions.

3 3.1 Algorithm Algorithm 1 Algorithm with Oracle ˆf number of edges (u, v) such that P [u] P [v] to additive error ɛdn if ˆf > 3 ɛdn then output fail and halt Choose S = O( 1 ɛ ) random nodes for s S do if P [s] K or P [s] is not H minor free then reject and halt. Accept The first part of the algorithm (the part before the choose statement), makes O( 1 ɛ ) calls to the oracle. The second part, makes O( d ɛ ) calls to the oracle, making the total number of calls O( d ɛ 3 ) 3. Analysis If G is H minor free: We know that if G is H minor free, then we will not halt on the first half of the algorithm because E[ ˆf] ɛdn 4 and with sampling bounds ˆf ɛdn 4 + ɛdn = 3 ɛdn. Since, our oracle produces H minor free graphs we can say s V, P [s] is H minor free. We see that our algorithm works when G is H minor free. Let us now analyze what happens when G is ɛ-far from H minor free. Case 1:P s output does not satisfy {(u, v) E : P (u) P (v)} < ɛdn. Using out sampling bounds, we can say that ˆf > ɛdn ɛdn = 3 ɛdn which means the algorithm will out put fail with probability Case : P s output does satisfy {(u, v) E : P (u) P (v)} < ɛdn. This means that G can be created from G with edges in C removed. It is important to note that if G is ɛ -close to G, This means that if G is ɛ-far from aving a property, then G is ɛ -far from having the property. Since G is ɛ ɛdn -far from H minor free, we much change edges to make it H minor free, which touch at least ɛn nodes. So with probability ɛ, we will pick a node in a component which is not H minor free. We now need to implement the partition oracle P. 4 Implementing Partion Oracle We will implement our partion oracle by first defining a global partitioning stragety that does not work in sublinear time and then figuring out how to implement the strategy locally. 4.1 Global Partitioning Strategy We will first define a useful concept known as an isolated neighborhood. 3

4 Definition S is (δ, k)-isolated neighborhood of node v if: v S S is connected S k Number edges connecting S and S is δ S. In hyperfinite graphs, most nodes have (δ, k)-isolated nodes. Why? Because if G is hyperfinite, then we know that there exists some partitioning. However, we need there to be a partitioning for the remaining graph since an algorithm may find a different partition. Luckily, we know that no matter what was removed, due to H minor free properties, we still have an H minor free graph that is still hyperfinite. Algorithm Global Partitioning Algorithm π 1,..., π n be nodes in a random order p = φ for i = 1..n do if π i still in graph then, if (δ, k)-isolated neighborhood of π i in remaining graph. then S=this neighborhood S = {π i } P = P {S} Remove S and adjacent edges from graph We know that if S is (δ, k)-isolated, then it will contribute δ s edges, which overall means δn. However, we cannot say the same if S = {π i } (one node). But we can show that we will not have too many of these singleton cases. Lemma 9 If G is a subgraph of a (hyperfinite) graph G such that G has δn nodes, then ɛ 30 fraction of nodes in G do not have (ɛ, k)-isolated neighborhoods. The idea for this proof is that since G is H minor free, then G is H minor free, meaning G is hyperfinite as well. This means there exists a partition such that most nodes in G are in (δ, k)- isolated neighborhoods. This means that with a high probability, a random π i will be in a (δ, k)-isolated neighborhood. 4. Local Simulation of Partitioning Oracle We will input v, assume access to a random function π(v) such that π : v [n] and then we will output P [v]. 4

5 Algorithm 3 Local Simulation for all w such that π(w) < π(v) and w is distance k from v do recursively compute P [w] if w such that v P [w] then P[v]=P[w] look for (δ, k)-isolated neighborhood of v (ignoring nodes which are in P [w] for smaller ranked ws) if find one then P [v]= neighborhood P [v] = {v} This algorithm makes d O(k) recursive calls. However, with the bounds we learned in the previous lecture, the runtime is do(k). The best possible runtime right now is d O(log ( 1 ɛ )). 5